Statistical computation for heat and mass transfers of water-based nanofluids containing Cu, Al2O3, and TiO2 nanoparticles over a curved surface

Nanofluid is a specially crafted fluid comprising a pure fluid with dispersed nanometer-sized particles. Incorporation these nanoparticles into pure fluid results in a fluid with improved thermal properties in comparison of pure fluid. The enhanced properties of nanofluids make them highly sought after, in diverse applications, consisting of coolant of devices, heat exchangers, and thermal solar systems. In this study hybrid nanofluid consisting of copper, alumina and titanium nanoparticles on a curved sheet has investigated with impact of chemical reactivity, magnetic field and Joule heating. The leading equations have converted to normal equations by using appropriate set of variables and has then evaluated by homotopy analysis method. The outcomes are shown through Figures and Tables and are discussed physically. It has revealed in this study that Cu-nanofluid flow has augmented velocity, temperature, and volume fraction distributions than those of Al2O3-nanofluid and TiO2-nanofluid. Also, the Cu-nanofluid flow has higher heat and mass transfer rates than those of Al2O3-nanofluid and TiO2-nanofluid.

understanding and harnessing the relationship between Brownian motion and thermophoresis are essential for optimizing heat transfer efficiency and designing advanced materials with tailored thermal properties 36 .
Joule heating, also known as ohmic heating, is a process where electrical energy is converted into heat when an electric current passes through a conductor with some resistance 37 .The phenomenon is named after James Prescott Joule, who first described it in the mid-nineteenth century.Joule heating is prevalent in various electrical devices, such as resistive heaters, electric stoves, and incandescent light bulbs, where the electrical energy is intentionally converted into heat for practical applications 38 .In some cases, particularly in electronic devices and integrated circuits, controlling and minimizing Joule heating is crucial to prevent overheating and ensure the efficient operation of the system.The impact of Joule heating on heat transfer is significant, particularly in electronic devices and conductive materials 39 .In electronic components, this phenomenon can influence the overall thermal management of the device.The heat produced due to Joule heating can lead to temperature gradients and thermal stresses, affecting the performance and reliability of the system 40 .Efficient heat dissipation strategies become crucial to prevent overheating and potential damage.Additionally, in applications like microelectronics and integrated circuits, understanding and managing Joule heating is essential for designing effective cooling systems, heat sinks, and thermal interfaces to enhance overall heat transfer and prevent thermal-induced failures in electronic components.Otman et al. 41 discussed mathematically the analysis of mixed convection stagnant point flow on an extending Riga surface using Joule heating impacts.Irfan et al. 42 inspected thermally on the depiction of mixed convective and radiative nanoparticles flow with Joule heating effects and has proved that thermal transportation has amplified with upsurge in Eckert number.Prakash et al. 43 discussed radiative and bio-convective nanoparticles flow of fluid on extending bi-directional sheet with impacts of Joule heating, modified diffusions and MHD effects.
Thermal convective conditions in fluid flow describe the dynamic process of heat transfer through the movement of a fluid, influenced by temperature gradients.Natural convection occurs when temperature differences induce buoyancy forces, causing the fluid to circulate spontaneously.Forced convection involves externally induced fluid motion, often using pumps or fans, to enhance heat transfer.Mixed convection combines aspects of both natural and forced convection, prevalent in scenarios where external forces and buoyancy both contribute to fluid motion 44 .These thermal convective conditions are pivotal in diverse applications, including the design of heat exchangers, cooling systems, and various industrial processes.The optimization of thermal convective systems requires a comprehensive understanding of fluid behavior, geometry, and external influences, achieved through computational modeling, experimentation, and theoretical analyses 45 .Hamza et al. 46 examined MHD time-dependent flow of fluid with convective constraints at the boundary and have noted that the velocity and thermal distributions have augmented with escalation in Biot number.Rashad et al. 47 inspected Williamson MHD nanofluid flow on a penetrable surface with convective constraints at the boundary and have proved that convective constraints at the boundary and upsurge in thermal radiations has escalated the thermal distribution and declined the velocity panels.Baag et al. 48studied free convective nanofluid flow on a stretching sheet using the impacts of heat source with heating convective constraints at the boundary.The impact of thermal convective conditions on heat transfer in fluid flow is profound, efficient heat exchange and thermal management in various engineering systems, such as heat exchangers or electronic devices; depend on thermal convective conditions 49 .Prasad et al. 50studied nanofluid couple stress flow with convective constraints at the boundary using temperature-based characteristics and impacts of MHD.
In this study hybrid nanofluid consisting of copper, alumina and titanium nanoparticles on a curved surface has investigated with impact of chemical reactivity, magnetic field and Joule heating.The leading equations have converted to normal equations by using appropriate set of variables.The purpose of this analysis is to address the subsequent research questions: • Among water-based nanofluids comprising of Cu, Al 2 O 3 and TiO 2 , which exhibits more pronounced velocity and temperature distributions due to inherent factors?• Among water-based nanofluids comprising of Cu, Al 2 O 3 and TiO 2 , which demonstrates higher surface drags and heat transfer rates due to inherent factors?• Between curved and flat surfaces, which surface type experiences predominant effects in terms of velocity and temperature distributions, surface drags, and heat transfer rates?

Problem statement
Assume the flow of nanofluids containing copper ( Cu ), alumina ( Al 2 O 3 ) and titanium dioxide ( TiO 2 ) nano- particles over a curved surface which is heated up with convective heating.The curved sheet stretches along primary direction, denoted by S , with velocity u w = aS such that a > 0 .A magnetic field of strength B 0 along R− direction is also taken into consideration.Furthermore, a hot working base fluid (water) having heat trans- fer coefficient h f is taken into consideration.The reference temperature is denoted by T f which is greater than the surface temperature T w (i.e.,T f > T w ).Also, the volumetric fraction distribution of the surface is denoted by C w and free stream is denoted by C ∞ as portrayed in Fig. 1.The subsequent conventions have been made to investigate the nanofluids flows: • The nanofluids flows are affected by the chemical reactivity, Joule heating, and magnetic field.
• The curved surface is taken to be impermeable.
• The curved surface is kept hot with a hot working fluid.
Using above stated assumptions we have [51][52][53] : The constraints at boundaries are 53 : The thermophysical features of the nanofluid are: Table 1 depicts the measured values of pure fluid and nanoparticles.
To convert the equations mentioned earlier, the similarity variables are established as follows: (1) www.nature.com/scientificreports/By utilizing the similarity variables mentioned above, Eq. ( 1) is obvious, and Eq. ( 2) converted to: where α = Ŵ + δ .Equation ( 3) can be reduced as: By taking the derivative of Eq. ( 10) with respect to Ŵ and is incorporating it into Eq.( 9), we derive: While rest of the main equations are converted as Pressure can be reduced as: 1 Pr Table 1.Measured values of the base fluid and nanoparticles in the experiment [54][55][56] .16), δ shows the curvature factor, M shows magnetic factor, Bi is thermal Biot number, Ec is Eckert number, γ signifies chemically reactive factor, Nb is Brownian motion parameter, Nt presents the thermophoresis factor, Sc designates Schmidt number and Pr shows Prandtl number.
Main quantities are depicted as: where Using Eq. ( 8) in Eq. ( 17) we have where Re S = aS 2 ν f is local Reynolds number.

Solution by HAM
For the application of HAM, we introduced the initial guesses and linear operators as follows: with properties: Above σ 1 − σ 8 are fixed values.For further applications of HAM, please see [57][58][59] .

Convergence analysis
This section presents the convergence analysis of the applied method called HAM.This method has a convergence control parameter which regulates the convergence of HAM. Figure 2 is exhibited to see the convergence area of velocity, temperature and volume fraction distributions.From Fig. 2, we confirm that the convergence of velocity takes place on interval −1.0 ≤ f ≤ 0.0 , convergence of thermal panels occur in −2.3 ≤ θ ≤ 0.25 , and volume fraction distribution converges in −2.7 ≤ φ ≤ 0.5.

Code confirmation
This section is presented to validate our outcomes with earlier available results.The obtained results are determined for − √ Re S C fS while varying δ and M = = 0 .The results are presented in Table 2. From this Table, we have confirmed that the results obtained for a special case are in good relation with those published results.

Results and discussion
This section offers the physical debate of the consequences attained during this analysis.In this study hybrid nanofluid consisting of copper, alumina and titanium nanoparticles on a curved surface has investigated with impact of chemical reactivity, magnetic field and Joule heating.The curved surface is kept hot with a hot working fluid.The curved surface is taken to be impermeable.The HAM method is used to calculate the results for different flow profiles via different embedded factors.The obtained results are exhibited in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14, and Tables 3 and 4. The impression of curvature factor ( δ ) on velocity profiles ( f ′ (Ŵ) ) of Al 2 O 3 -nanofluid, Cu-nanofluid, and TiO 2 -nanofluid is shown in Fig. 3. Higher δ increases the velocity profiles of all three types of fluids.The reason is that the increasing δ means that the curvature of the surface readuces or increases the radius of curvature which makes the surafce flatten.As the surface become flatten then the ( 16) L f σ 1 + σ 2 Ŵ + σ 3 e −Ŵ + σ 4 e Ŵ = 0, L θ σ 5 e −Ŵ + σ 6 e Ŵ = 0, L φ σ 7 e −Ŵ + σ 8 e Ŵ = 0, opposing force at the flatten surface lowers which results higher velocity.The obtained results are investigated in the existence of magnetic field which means that the electrical conductivities of the different nanomaterials (i.e., Al 2 O 3 , Cu, and TiO 2 ) has significant role here.From the obtained results, we can see that the Cu-nanofluid has greater velocity than those of Al 2 O 3 and TiO 2 -nanofluids.The impact δ on temperature profiles ( θ(Ŵ) ) of Al 2 O 3 -nanofluid, Cu-nanofluid, and TiO 2 -nanofluid is portrayed in Fig. 4.An expansion in values of δ increases the temperature profiles of Al 2 O 3 -nanofluid, Cu-nanofluid, and TiO 2 -nanofluid.Physically, the curvature factor and kinemetic viscosity are inversually proportional to each other which means that the higher curvature factor reduces the kinematic viscosity which results enhancement in θ(Ŵ) .Hence, the higher δ increases θ(Ŵ) .Since from Table 1, we know that the thermal conductivities of the different nanomaterials (i.e., Al 2 O 3 , Cu, and TiO 2 ) has significant role here.From the obtained results, we can see that the Cu-nanofluid has greater thermal conductivity than those of Al 2 O 3 and TiO 2 -nanofluids.Therefore, Cu-nanofluid has greater thermal distribution than those of Al 2 O 3 and TiO 2 -nanofluids.The impact of M on f ′ (Ŵ) for Al 2 O 3 -nanofluid, Cu-nanofluid, and      electrical conductivity than those of Al 2 O 3 and TiO 2 -nanofluids.Therefore, the Cu-nanofluid has greater velocity than those of Al 2 O 3 and TiO 2 -nanofluids.The impact of M on θ (Ŵ) of the Al 2 O 3 -nanofluid, Cu-nanofluid, and TiO 2 -nanofluid is exposed in Fig. 6.The greater values of M increases the temperature profiles of Al 2 O 3 -nanofluid, Cu-nanofluid, and TiO 2 -nanofluid.As the magnetic strength intensifies, the friction force on the sheet surface rises, leading to a higher rate of thermal transference attributed to increased friction.The higher rate of heat   transfer increases the temperature profiles of the nanofluids.Since, Cu-nanofluid has greater electrical conductivity so it will have greater resistive force and rate of heat transfer as well.Therefore, Cu-nanofluid has greater thermal distribution than those of Al 2 O 3 and TiO 2 -nanofluids.The impression Ec on thermal profiles for Al 2 O 3 -nanofluid, Cu-nanofluid, and TiO 2 -nanofluid is shown in Fig. 7. Greater Ec is responsible for upsurge in θ(Ŵ) for Al 2 O 3 -nanofluid, Cu-nanofluid and TiO 2 -nanofluid.The higher Ec transmits the kinetic energy into

Conclusion
In this study hybrid nanofluid consisting of copper, alumina and titanium nanoparticles on a curved sheet has investigated with impact of chemical reactivity, magnetic field and Joule heating.The leading equations have converted to normal equations by using appropriate set of variables and has then evaluated by homotopy analysis method (HAM).The results are shown through Figures and Tables and are discussed physically.The ultimate results of the current analysis are: • The velocity and temperature panels augmenting functions of curvature factor.
• Growing values of magnetic factor reduce the velocity profiles while opposite impact is found for the tem- perature distribution.

Figure 1 .
Figure 1.Graphical display of flow problem.

•
Brownian motion factor, thermophoresis factor and thermal Biot, Eckert numbers have direct relations with the thermal distribution.• The Brownian motion, chemical reactivity factors, and Schmidt number has inverse relation with the volume fraction distribution.• The Cu-nanofluid flow has higher velocity, temperature, and volume fraction distributions than those of Al 2 O 3 -nanofluid and TiO 2 -nanofluid.• The Cu-nanofluid flow has higher heat and mass transference rates than those of Al 2 O 3 -nanofluid and TiO 2 -nanofluid.

Table 2 .
Comparison of − √ Re S C fS for different values of δ with M = = 0.δ Rosca

Table 3
Al 2 O 3 -nanofluid, Cu-nanofluid, and TiO 2 -nanofluid) via , M , Ec , Nb , Nt and Bi .From the obtained results, we have found that Nu s Ec , Bi and Nt while reduces via Nb .Physically, the higher values of increase the thermal conductivities of the nanofluids which results higher rate of heat transfers.Comparing the three different nanofluids (i.e., Al 2 O 3 -nanofluid, Cu-nanofluid, and TiO 2 -nanofluid), the Cu-nanofluid flow has higher volume fraction distributions than those of Al 2 O 3 -nanofluid and TiO 2 -nanofluid.Table4shows the variation in mass transfer rate Nu s shows the variation in heat transfer rate Nu s √ Re s of the three different nanofluids (i.e.,

Table 3 .
Impacts of M , Ec , Nb , Nt , Bi and on Nu S

Table 4 .
Sherwood number for variations in Nb and Nt.www.nature.com/scientificreports/ of the three different nanofluids (i.e., Al 2 O 3 -nanofluid, Cu-nanofluid, and TiO 2 -nanofluid) via Nb , Nt and Sc .From the obtained results, we havefound that Nu s √ Re s increases via Nb and Nt while reduces via Sc.